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conclusion, on the supposition that the subject of the conclusion is comprehended in that of the major premise. This supposition the minor premises affirm.

Syllogisms complete the development of thought in all its essential elements and relations. Terms represent ideas as they stand related to single objects taken individually or collectively; propositions represent ideas as they stand related to objects considered as subjects and predicates; and syllogisms represent the objects of the ideas expressed by propositions in their relations to each other as the subjects of premises and conclusions. The development of thought is commenced by terms; extended by propositions; and completed by syllogisms. There is no element of thought in discourses, which is not found in syllogisms

CHAPTER XIX.

MATHEMATICAL REASONING.

§ 259. Synthetical reasoning may be divided into two varieties, corresponding to the two categories of quantity and reality. That which belongs to the category of quantity is beautifully exemplified in the sciences of pure and mixed mathematics. It is used to a greater or less extent in all the sciences, and in all departments of knowledge; but in pure mathematics it is used almost exclusively, and most of its principles may be conveniently discussed under the title of mathematical reasoning.

The object of mathematical reasoning is quantity. Quantity denotes any thing which can be measured or estimated as one or more, and comprehends all the objects of our knowledge, considered with respect to those properties which we discover by the categorical exercises of judgment belonging to quantity. Bodies, spirits, thoughts, affections, actions, space, and time, are all quantities. Quantities are of two kinds, pure and mixed. Pure quantities consists of simple elements which are perfectly uniform, and perfectly ideal, and which cannot be conceived of except in terms of elements considered as one or more, and greater or less. Of this discription are time and space. Mixed quantities

are objects which possess other properties besides those of quantity; and require to be estimated not only in terms of quantity, but in those of causality and dependence. Of this description are bodies, spirits, thoughts, affections, and

actions.

§ 260. Pure quantities are estimated only in terms denoting their elements; and denoting combinations of those elements, in definite modes. Thus times are estimated in terms of times; superficial extent, in terms of superficial extents; and solid portions of space, in terms of solids. In the same manner, time is estimated in terms of seconds, minutes, days, years, and centuries.

But, in conceiving of mixed quantities, we regard them not as quantities merely, but as causes, states, operations, and effects; and apply to them the categories of quantity, in subordination to those of causality and dependence; so that in these estimates, quantity constitutes a limitation of causality and dependence, in respect to space and time. The extension of a body in space, is the extension of that body as a cause, or a state or operation of some cause. duration of a body, and of a state or operation of a body in time, is the period of time during which that body and that state or operation of a body continues.

The

§ 261. The principal branches of Mathematics are Arithmetic, Algebra, Geometry, Trigonometry, Conic Sections, and the Differential and Integral Calculus.

Arithmetic relates to quantity as denoted by numbers, and considered simply as consisting of things of the same genus; or as of higher or lower orders of the same genus. Algebra relates to quantity, considered as denoted by letters, some of which are used as definite symbols of quantity, and others as indefinite. Geometry relates to magnitudes, embracing lines, surfaces, and solids, which are represented by lines, surfaces, and solids. Trigonometry relates to triangles; Conic Sections, to ellipses, hyperbolas, and parabolas, figures which are capable of being generated by sections of cones, all of which may be represented by plain figures, or by letters denoting the essential relations of their different parts to each other. The Differential and Integral Calculus relate to many different kinds of quantities, which they investigate by means of indefinitely small elements of them, denoted by letters.

Every branch of mathematical reasoning is prosecuted chiefly by means of symbols. In Arithmetic, the symbols of quantity are all definite; in Algebra, and other higher branches, they are part definite, and part indefinite. The end of all mathematical reasoning is from given quantities to determine other quantities, and from known truths to deduce unknown; and the whole science may be reduced to propositions and problems. Propositions denote conclusions to be proved, and problems, operations to be performed. The following are examples of propositions: Any one side of a triangle is less than the sum of the other two; a diameter divides a circle and its circumference into two equal parts. The following are examples of problems: To divide a straight line into two equal parts; to find the center of a given circle; to inscribe a circle in a triangle.

§ 262. The use of symbols in mathematical reasoning is worthy of particular remark. The purpose which they answer is to represent quantities, operations, and judgments, and to enable us to keep them steadily in view. Definite symbols represent quantities definitely; indefinite ones, indefinitely; but both represent them and represent them truly, as far as they are understood. The language of mathematics is superior to that of general reasoning, in respect to the clear discrimination of known quantities from unknown, and defined quantities from indefinite. This discrimination can be made in all departments of reasoning, and is necessary to correct and successful reasoning; but it is not as easily made in other departments of investigation, as in that of quantity. Another advantage possessed by the symbols of quantity over words, is the precision with which they are used, and the simplicity of the objects which they denote. A line is a thing of which we have a perfect conception, and which we cannot confound with a plane or with a solid. The same is true of other figures. But it is not so with causes, and states of being. A body may be fully grasped, and distinctly and definitely conceived of, in respect to its figure and size, but not in respect to its causality and dependence, or the possible states of being through which it is capable of passing, and in which it is capable of existing. Another circumstance which makes the symbols of mathematical reasoning far superior to the language of common discourse as an instrument of reasoning, is their extreme conciseness. The advantage of this is greater than can

easily be imagined. Some apprehension of it may be obtained from considering what impediments would be thrown in the way of adding, subtracting, multiplying, and dividing in Arithmetic, if we were required to substitute words for figures, and to perform these processes by the use of words as symbols of quantities. The system of figures in arithmetic and that letters and other symbols in the higher branches of mathematics, are constituted on principles of generalization which adapt them to express quantities in all their conceivable relations in the most perfect manner; but the expression of quantities in their known and conceivable relations, is the expression of all the possible deductions of reason respecting them.

§ 263. The primary conditions of all mathematical reasoning are elementary ideas of quantity, such as units, lines, planes, and solids. We cannot reason without something to reason from, and to make the subject of our judgment; but having subjects concerning which to form judgments, we can form numerous judgments respecting them. The elementary ideas of quantity are obtained contemporaneously with perceptions of material objects, and are common to the entire human race. These ideas are expressed and communicated by common names and definitions. The elementary idea which is the basis of arithmetic is number, and that which is the basis of geometry is magnitude.

Every department of mathematical reasoning commences with conceptions expressed by definitions, and is extended from those primary conceptions by consecutive judgments. In arithmetic we commence with certain conceptions of numbers, and with a system of figures adapted to express numbers. From these elementary ideas we deduce rules of addition, subtraction, multiplication, and division, and all the other rules and principles of this science. These deductions are made consecutively, from our primitive ideas given us by definition, or from other ideas derived directly or remotely from them. From the fact that the system of numbers consists of a classification of figures according to their places, making each figure represent units in the right hand place, tens in the second place from the right hand, hundreds in the third, and so on; we infer that in order to be added correctly, they must be written down in columns, with units under units, tens under tens, and so on; we also infer, that in expressing their sum underneath, we shall

proceed correctly by writing down under each column the right hand figure of the number expressing the sum of that column, and adding the left hand figure or figures of that sum, in cases where it is expressed by a plurality of figures, to the next column; and so on. These conclusions may be obtained in the first instance, from examples illustrating them in particular cases, and subsequently from the principles involved in the theory of numbers.

Most of the judgments which constitute a series of deductions in mathematical reasoning, are derived from comparisons of two quantities with each other; and are conclusions that the quantities are equal or unequal, or that they sustain other relations which are discoverable by such comparisons. From the two quantities compared, another quantity is inferred. This last discovered quantity is compared with some pre-discovered one, and another still discovered, and so on indefinitely.

§ 264. In mathematical reasoning we make no use of major and minor premises, because our conclusions are not deductions from general ideas respecting things comprehended in the subjects of those ideas, but from particular ideas respecting the yet undiscovered properties and relations of their objects. The attempt to reduce mathematical reasoning to the syllogistic form, and to resolve it on the same principle, is one of the most remarkable aberrations of modern science.

The same habits of generalization by which we reduce our ideas of other objects to general ideas, lead us to make a similar reduction of our ideas of quantity. The ideas under which these are comprehended, are called axioms. Thus we reduce all our judgments of the equality of the parts of quantities to wholes, under the axiom; that a whole is equal to the sum of all its parts; and all our judgments, that things equal to the same thing are equal to each other, under the corresponding axiom; that things equal to the same thing are equal to each other. All other general ideas of the relations of quantities have a similar origin. But these general ideas, which are of the greatest use in other departments of reasoning, are of very little use in reasoning on quantity. With the four parts of an apple before us, we infer their equality to the whole, without any assistance from the axiom, that a whole is equal to the sum of all its parts; and on measuring two objects with a rule, and find

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